# Formula for Angle between Two Lines

Here, you will learn formula for angle between two lines, equation of straight line making an angle with a given line and reflection and image of a point in a line and also length of perpendicular from a point on a line.

Let’s begin –

## Angle between Two Lines

(a)  If $$\theta$$ be the angle between two lines : y = $$m_1x + c_1$$ and y = $$m_2x + c_2$$, then

tan$$\theta$$ = $$\pm$$ ($$m_1-m_2\over {1+m_1m_2}$$)

(b)  If the equation of lines are $$a_1x + b_1y + c_1$$ = 0 and $$a_2x + b_2y + c_2$$ = 0, then these lines are –

(i)  Parallel  $$\iff$$   $$a_1\over a_2$$ = $$b_1\over b_2$$ $$\ne$$ $$c_1\over c_2$$

(ii)  Perpendicular  $$\iff$$  $$a_1a_2$$ + $$b_1b_2$$ = 0

(iii)  Coincident  $$\iff$$  $$a_1\over a_2$$ = $$b_1\over b_2$$ = $$c_1\over c_2$$

Equation of Straight line making an angle with a Line :

Equation of line passing through a point ($$x_1,y_1$$) and making an angle $$\alpha$$, with the line y = mx + c is written as

y – $$y_1$$ = $$m \pm tan\alpha\over {1 \mp mtan\alpha}$$(x – $$x_1$$)

## Reflection and image of a point in a line :

Let P(x,y) be any point, then its image with respect to

(a)  x-axis is Q(x,-y)

(b)  y-axis is R(-x, y)

(c)  origin is S(-x,-y)

(d)  line y = x is T(y,x)

(e)  Reflection of a point about any arbitrary line : The image (h,k) of a point P($$x_1,y_1$$) about the line ax+by+c = 0 is given by following formula.

$$h-x_1\over a$$ = $$k-y_1\over b$$ = -2($$ax_1+by_1+c\over {a^2+b^2}$$)

and the foot of perpendicular (p,q) from a point ($$x_1,y_1$$) on the line ax+by+c = 0 is given by following formula.

$$p-x_1\over a$$ = $$q-y_1\over b$$ = -($$ax_1+by_1+c\over {a^2+b^2}$$)

## Length of perpendicular from a point on a line :

Length of perpendicular from a point ($$x_1,y_1$$) on the line ax + by + c = 0 is

|$$ax_1 + by_1 + c\over {\sqrt{a^2+b^2}}$$|

In particular, the length of perpendicular from the origin on the line ax + by + c = 0 is

P = $$|c|\over {\sqrt{a^2+b^2}}$$.

Hope you learnt formula for angle between two lines and all other concepts. Practice more questions to learn more and get ahead in competition. Good Luck!